I have not been able to provide a copy of my SIN yet to my new employer because I am waiting for my health card to come in so I can use that as ID to even get the SIN. They say they want it by Friday but I wont have my health card yet by then. Can they fire me for this?

Yes, your employer could terminate your employment, and could attempt to allege frustration of contract for failing to provide the required documents to legally work. If you do get terminated, review your potential entitlements using the Severance Calculator and then give our office a call for a consult.